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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Homological dimensions of stable homotopy modules and their geometric characterizations


Author: T. Y. Lin
Journal: Trans. Amer. Math. Soc. 172 (1972), 473-490
MSC: Primary 55E45
DOI: https://doi.org/10.1090/S0002-9947-1972-0380789-8
MathSciNet review: 0380789
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Abstract: Projective dimensions of modules over the stable homotopy ring are shown to be either 0, 1 or $\infty$; weak dimensions are shown to be 0 or $\infty$. Also geometric charactetizations are obtained for projective dimensions 0, 1 and weak dimension 0. The geometric characterizations are interesting; for projective modules they are about the cohomology of geometric realization; while for flat modules they are about homology. This shows that the algebraic duality between “projective” and “flat” is strongly connected with the topological duality between “cohomology” and “homology". Finally, all the homological numerical invariants of the stable homotopy ring—the so-called finitistic dimensions—are completely computed except the one on injective dimension.


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Keywords: Stable homotopy ring, stable homotopy module, finitistic dimensions, projective dimension, weak dimension, injective dimension, higher order homology operation, cohomology operation, Eilenberg-Mac Lane spectrum, Postnikov system, projective module, flat module, injective module, Puppe sequence, mapping cone sequence, spectral sequence, Hurewicz homomorphism
Article copyright: © Copyright 1972 American Mathematical Society